Several learning methods, such as Expected Patch Log Likelihood (EPLL) [Zoran and Weiss, 2011], patch prior guided internal clustering with low rank (PCLR) [Chen et al., 2015], and Patch Group Prior based Denoising (PGPD) [Xu et al., 2015b] were proposed to derive priors from natural noise-free images. However, these learned priors are generic for natural images and are not specific to any image category. Similarly, many early works [Elad and Aharon, 2006; Mairal et al., 2009; Dong et al., 2011] learn an over-complete dictionary of image patches from an external noise-free database and impose non-local self-similarity through a sparse representation. Below are some of the prominent learning methods.
2.2.4.1 Denoising via Singular Value Decomposition
[Elad and Aharon, 2006] proposed a dictionary learning algorithm based on a sin- gular value decomposition (SVD) called K-SVD. It is a simple iterative algorithm and learns from the noisy image itself. Each iteration is composed of two steps: i) learn the coefficients using orthogonal matching pursuit (OMP) [Chen et al., 1989; Pati et al., 1993] (algorithm selects the dictionary atoms sequentially) in each path, ii) update one column of the dictionary at a time. Usually, a few iterations are sufficient to achieve good results. Denoising is performed patch-wise and then inserted back into its original location. Moreover, averaging is conducted in areas of overlapping patches.
2.2.4.2 Non-local Sparse Models
Similar to KSVD, Non-local sparse model (NLSM) also learns from the dictionary from the noisy image. However, the difference is that NLSM applies sparsity on the learned patches. The underlying idea is that similar noisy patches can be ap- proximated using the same sparse decomposition. The purpose of looking for the similar patches are also exploited by many internal denoising algorithms [Dabov et al., 2007b; Buades et al., 2005]. NLSM visual results are considered to be the best
in the current lot such as BM3D, NLM etc. However, its computational cost is very high.
2.2.4.3 Spatially Adaptive Iterative Singular Value Thresholding
[Dong et al., 2013] proposed an`p,qnorm constraint to promote patch similarity and derived a denoising solution via spatially adaptive iterative singular-value threshold- ing (SAIST).
argmin A
||UA−y||2
2+α||A||p,q, (2.42)
whereUis the learned dictionary andAare the collection of sparse coefficients and
UA represents the clean image i.e. x=UA. The ||A||p,q is defined by [Cotter et al., 2005] as ||A||p,q= N
∑
i=1 ||γi||pq, (2.43)where γi is the i-th row of the matrix A. This algorithm achieved sparse represen-
tation using clustering and is formulated by combining the strengths of dictionary learning and structural clustering. In other words, SAIST employs singular value de- composition to represent image patches sparsely and then iteratively remove noise by thresholding the singular values using BayesShrink [Chang et al., 2000]. This method is not only applicable to denoising but other task as well such as image completion. SAIST is computationally expensive as it requires ten iterations to produce the final denoised image.
2.2.4.4 Gaussian Mixture Model priors
As an alternative, [Zoran and Weiss, 2011] aims to learn a statistical prior of natural image patches, such as the Gaussian Mixture Model (GMM) of natural image patches or patch groups for patch reconstruction in a maximum likelihood framework. Ex- pected Patch Log Likelihood (EPLL) contrasts itself than other denoising algorithms by taking a posteriori approach. This method is already described in section 2.1.2.3 as it is applicable for deblurring and denoising with a small difference in formulation. For denoising, the Equation 2.33 becomes
argmin x
||x−y||22+
∑
i
logp(Pix). (2.44)
EPLL uses half-quadratic splitting optimization which introduces auxiliary vari- ables. The process proceeds with alternation between two phases: i) fixing the image
patches while updating auxiliary variables, ii) keeping the auxiliary variables con- stant and updating the image patches. According to [Zoran and Weiss, 2011], this process is iterated for four to five times. The performance of EPLL is comparable to BM3D and NLSM.
Recently, several authors adapted [Zoran and Weiss, 2011]’s patch prior repre- senting image-specific and class-specific semantics. Based on a Gaussian Mixture Model (GMM), this generic prior captures statistics of natural patches by perform- ing the Expectation-Maximization (EM) algorithm on a large dataset of clean patches [Zoran and Weiss, 2011; F. Chen and Yu, 2015; Xu et al., 2015a].
[Chen et al., 2015] proposed internal clustering guided by external patches and apply low-rank decomposition and termed it as patch prior guided internal cluster- ing with low rank (PCLR). Low-rank regularization is applied on similar internal patches clustered using global similarity in the noisy image rather than local block matching. Subsequently, the method learns Gaussian Mixture Model (GMM) prior to guide the patch clustering and perform a low-rank subspace learning. Such a grouping based low-rank regularization makes the underlying patch restoration very robust to noise. The performance of PCLR is marginally better then BM3D.
Patch Group Prior based Denoising (PGPD) is introduced by [Xu et al., 2015b], which uses a patch group to denoise the noisy patches. After subtraction of mean from the patch group, it represents the non-local self-similar prior to natural images. Then a GMM is learned from the non-local self-similar patch group extracted from natural images. Lastly, sparse coding is applied to the patch group for efficiency. The denoising results of PGPD is below than most state of the art algorithms; however, its efficacy is comparable to BM3D [Xu et al., 2015b].
[Teodoro et al., 2016] proposed an approach to locally adapt the GMM prior [Zo- ran and Weiss, 2011] to the class of each patch. This method enables patch-based image enhancement for multiple classes appearing in the same image. The authors employ segmentation to differentiate between different classes present in the image; however, this approach may result in degraded denoised outputs as image segmen- tation itself is a difficult task especially in the presence of noise.
2.2.4.5 Adaptive Image Denoising
AID is an acronym for Adaptive Image Denoising [Luo et al., 2016]. This work is aimed to adapt the generic patch prior to one that is specific to the patch statistics of the input image. The core of the method is a modified version of the Expectation- Minimization (EM) algorithm on the noisy image or its pre-filtered version with an estimate of the noise. The results of this method are superior to the ones when there is no EM adaptation. Also the quantitative results for the proposed image denoising
algorithm yield better results than some state of the art algorithms mentioned earlier.